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We provide a method for obtaining upper estimates of the resolvent kernel ofthe Laplacian on a post-critically finite self-similar fractal that relies on aself-similar series decomposition of the resolvent. Decay estimates on thepositive real axis are proved by analyzing functions satisfying an interioreigenfunction condition with positive eigenvalue. These lead to estimates onthe complement of the negative real axis via the Phragmen-Lindelof theorem.Applications are given to kernels for functions of the Laplacian, including theheat kernel, and to proving the existence of a self-similar seriesdecomposition for the Laplacian resolvent on fractal blowups.

Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups